Untersuchungen über die Grundlagen der Mengenlehre I.

Zermelo, E. (1908). Untersuchungen über die Grundlagen der Mengenlehre I. Mathematische Annalen, 65(2), 261-281.

Type

Journal article

Intellectual base

Primary research

Year

1908

Volume

65

Issue

2

Pages

261-281

DOI

10.1007/bf01449999

DOI registrant

Springer Nature

Link

https://doi.org/10.1007/BF01449999

Cited from

mathematics, philosophy

Cited by

6 citations across 6 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Infinity
  • Subsequent foundational work — Zermelo's (1908) axiomatization

    This sourceFoundational axiomatization of set theory; axiom of choice and well-ordering principle.

    Supported in partVerified against the publisher's abstract

    Zermelo's own 1908 paper on the foundations of set theory confirms the work and author-year cited, but its readable text does not mention the axiomatization or the 'subsequent' framing.

    “beruht lediglich auf der Dedekindschen Kettentheorie (Was sind und was sollen die Zahlen? § 4) und vermeidet im Gegensatz zu den älteren Beweisen von E. Bernstein, sowie zu dem letzten Beweise von J.”

    Read by an automated reader; not a human review. How support was checked

• Paradox
  • T3: Set-Theoretic Foundations and Axiom Choice

    This sourceFoundational axiomatization of set theory; axiom of choice and well-ordering principle.

• Set and Membership
  • In classical (ZFC) set theory

    This sourceFoundational axiomatization of set theory; axiom of choice and well-ordering principle.

• Tolerance Paradox
  • The resolution is precisely a meta-level exception: Zermelo–Fraenkel replaces unrestricted comprehension with restricted (separation) comprehension, which only forms {x ∈ A : P(x)} — subsets of an already-existing set A.

    This sourceIntroduces the axiom of separation (restricted comprehension), blocking Russell's paradox while preserving ordinary set construction.

• Well-Foundedness (Well-Ordering)
  • … ring is Noetherian) is a core algebraic-geometry foundational result that depends on this; Zorn's lemma (equivalent to AC) supplies well-founded-style maximal elements in inductively-ordered sets; regular cardinals and stationary sets in set theory rely on well-foundedness arguments for their existence and uniqueness.

    This sourceAxiomatisation of set theory; precursor of the foundation axiom.

Domain-specific

• Zermelo set theory
  • Zermelo’s 1908 system formalizes sets, membership, and possibly urelements through a specified axiom list designed to block unrestricted comprehension while supporting classical analysis and well-ordering arguments.

Verification

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